On the convex structure of process positive operator valued measures

Abstract

Measurements on quantum channels are described by so-called process operator valued measures, or process POVMs. We study implementing schemes of extremal process POVMs. As it turns out, the corresponding measurement must satisfy certain extremality property, which is stronger that the usual extremality given by the convex structure. This property motivates the introduction and investigation of the A-convex structure of POVMs, which generalizes both the usual convex and C*-convex structure. We show that extremal points and faces of the set of process POVMs are closely related to A-extremal points and A-faces of POVMs, for a certain subalgebra A. We give a characterization of A-extremal and A-pure POVMs in the Appendix. Corollary 3. M ∈ M(H, n) is A-extremal if and only if whenever MProof of Lemma 2. Here we use similar techniques as in [7]. We first prove the assertion for M = X 1 N 1 X 1 + X 2 N 2 X 2 , with X 1 , X 2 ≥ 0. If both X 1 and X 2 are invertible, N 1 , N 2 ∈ F by the definition of an A-face. So suppose that, say, X 1 is not invertible. For any λ ∈ (0, 1), M = X 1 N 1 X 1 + (λX 2 )N 2 (λX 2 ) + ( 1 − λ 2 X 2 )N 2 ( 1 − λ 2 X 2 ).